Transfer of complex structures and topological character of holomorphy

The following question by V. I. Arnold is answered in affirmative. Let $X,Y$, and $Z$ be three complex manifolds of the same dimension, $p\colon X\to Y$ the universal covering, $g\colon Y\to Z$ a nondegenerate holomorphic mapping. Suppose that, in the chain $X\overset p\to Y\overset g\to Z$, the term $Y$ is forgotten, while the complex structures on $X$ and $Z$ are changed so that the mapping $g\circ p$ remains holomorphic. Can one recover the forgotten term $Y$? Bibl. – 2 titles.